Real bounds, ergodicity and negative Schwarzian for multimodal maps

Strien, Sebastian van; Vargas, Edson

doi:10.1090/s0894-0347-04-00463-1

Cited by 90 publications

(63 citation statements)

References 21 publications

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“…Historically, the Koebe Principle was proven under the assumption that T has negative Schwarzian derivative, see [30,Chapter III.6]. A recent work of Kozlovski [23], extended to the multimodal case by van Strien and Vargas [42], shows that the C 2 assumption suffices. The magnitude of K = K(δ) = K 0 ((1 + δ)/δ) 2 , where K 0 depends only on the map.…”

Section: Statistics Of Return Times For Non-hyperbolic Interval Mapsmentioning

confidence: 99%

Return time statistics via inducing

Bruin¹,

Saussol²,

Troubetzkoy³

et al. 2003

Ergod. Th. Dynam. Sys.

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Section: Statistics Of Return Times For Non-hyperbolic Interval Mapsmentioning

confidence: 99%

Return time statistics via inducing

Bruin¹,

Saussol²,

Troubetzkoy³

et al. 2003

Ergod. Th. Dynam. Sys.

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“…The first results in this direction were obtained for circle diffeomorphisms in the 1920's by Denjoy [10], for critical circle maps in [66] and for circle maps with plateaus in [40]. For interval maps there are results, in increasing generality, [4,18,34,41,46,50,59]. On the other hand, interval exchange transformations can have wandering intervals, see e.g.…”

Section: Informal Summary Of the The Results In This Papermentioning

confidence: 99%

Asymmetric Unimodal Maps with Non-universal Period-Doubling Scaling Laws

Kozlovski

Strien

2020

Commun. Math. Phys.

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We consider a family of strongly-asymmetric unimodal maps $$\{f_t\}_{t\in [0,1]}$$ { f t } t ∈ [ 0 , 1 ] of the form $$f_t=t\cdot f$$ f t = t · f where $$f:[0,1]\rightarrow [0,1]$$ f : [ 0 , 1 ] → [ 0 , 1 ] is unimodal, $$f(0)=f(1)=0$$ f ( 0 ) = f ( 1 ) = 0 , $$f(c)=1$$ f ( c ) = 1 is of the form and $$\begin{aligned} f(x)=\left\{ \begin{array}{ll} 1-K_-|x-c|+o(|x-c|)&{} \text{ for } x<c, \\ 1-K_+|x-c|^\beta + o(|x-c|^\beta ) &{} \text{ for } x>c, \end{array}\right. \end{aligned}$$ f ( x ) = 1 - K - | x - c | + o ( | x - c | ) for x < c , 1 - K + | x - c | β + o ( | x - c | β ) for x > c , where we assume that $$\beta >1$$ β > 1 . We show that such a family contains a Feigenbaum–Coullet–Tresser $$2^\infty $$ 2 ∞ map, and develop a renormalization theory for these maps. The scalings of the renormalization intervals of the $$2^\infty $$ 2 ∞ map turn out to be super-exponential and non-universal (i.e. to depend on the map) and the scaling-law is different for odd and even steps of the renormalization. The conjugacy between the attracting Cantor sets of two such maps is smooth if and only if some invariant is satisfied. We also show that the Feigenbaum–Coullet–Tresser map does not have wandering intervals, but surprisingly we were only able to prove this using our rather detailed scaling results.

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“…A feasible candidate for g may be the map from [13,Theorem 7], whose wild attractor, similarly to C, is semiconjugate to the dyadic adding machine. Note that here things are further complicated by the fact that wild attractors have zero measure [39]. In the second case the situation should be simpler: we just need a map h possessing a wild attractor which is also an adding machine for h. In this regard [11] may be helpful.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

An Almost Everywhere Version of Smítal’s Order–chaos Dichotomy for Interval Maps

Blaya¹,

López²

2008

J. Aust. Math. Soc.

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We prove that if f : I = [0, 1] → I is a C 3 -map with negative Schwarzian derivative, nonflat critical points and without wild attractors, then exactly one of the following alternatives must occur: (i) R( f ) has full Lebesgue measure λ; (ii) both S( f ) and Scramb( f ) have positive measure. Here R( f ), S( f ), and Scramb( f ) respectively stand for the set of approximately periodic points of f , the set of sensitive points to the initial conditions of f , and the two-dimensional set of points (x, y) such that {x, y} is a scrambled set for f . Also, we show that if f is piecewise monotone and has no wandering intervals, then either λ(R( f )) = 1 or λ(S( f )) > 0, and provide examples of maps g, h of this type satisfying S(g) = S(h) = I such that, on the one hand, λ(R(g)) = 0 and λ 2 (Scramb(g)) = 0, and, on the other hand, λ(R(h)) = 1.2000 Mathematics subject classification: primary 37E05; secondary 37C05, 37D45, 37E15.

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Real bounds, ergodicity and negative Schwarzian for multimodal maps

Cited by 90 publications

References 21 publications

Return time statistics via inducing

Return time statistics via inducing

Asymmetric Unimodal Maps with Non-universal Period-Doubling Scaling Laws

An Almost Everywhere Version of Smítal’s Order–chaos Dichotomy for Interval Maps

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